UNIVERSITY OF CALIFORNIA, LOS ANGELES
Department of Economics
Economics 143 (Cameron) - Applied Regression
Analysis
Midterm Exhibits
|_* Hypothetical data on monthly demand for luxury American-made
|_* automobiles at a random sample of dealerships in different
|_* neighborhoods. Data include:
|_* age = median age of population in same zipcode as dealership
|_* inc = median household income of population in same zipcode
|_* cars = average number of American-made luxury cars sold per month
|_* price = median price of luxury cars stocked at dealership
|_* dist = proximity to nearst foreign luxury car dealership
|_sample 1 17
|_* Descriptive Statistics
|_stat / pcor
NAME N MEAN ST. DEV VARIANCE MINIMUM MAXIMUM
AGE 17 55.041 13.538 183.29 29.714 70.265
INC 17 186.94 38.230 1461.6 100.00 250.00
CARS 17 6.2659 3.1415 9.8688 1.7955 12.357
PRICE 17 53.941 2.5365 6.4338 49.000 59.000
DIST 17 11.414 6.5447 42.833 2.0830 21.814
CORRELATION MATRIX OF VARIABLES - 17 OBSERVATIONS
AGE 1.0000
INC 0.58242 1.0000
CARS 0.71033 0.67463 1.0000
PRICE 0.18276 0.31771 0.34422 1.0000
DIST 0.23821E-01 0.21740 -0.63699E-01 0.36003E-01 1.0000
AGE INC CARS PRICE DIST
|_* Regression 1
|_ols age inc cars price dist
R-SQUARE = 0.5316 R-SQUARE ADJUSTED = 0.3755
VARIANCE OF THE ESTIMATE-SIGMA**2 = 114.46
STANDARD ERROR OF THE ESTIMATE-SIGMA = 10.699
SUM OF SQUARED ERRORS-SSE= 1373.5
MEAN OF DEPENDENT VARIABLE = 55.041
LOG OF THE LIKELIHOOD FUNCTION = -61.4534
ANALYSIS OF VARIANCE - FROM MEAN
SS DF MS F
REGRESSION 1559.1 4. 389.77 3.405
ERROR 1373.5 12. 114.46 P-VALUE
TOTAL 2932.6 16. 183.29 0.044
VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 12 DF P-VALUE CORR. COEFFICIENT AT MEANS
INC 0.67960E-01 0.1019 0.6668 0.518 0.189 0.1919 0.2308
CARS 2.6439 1.227 2.155 0.052 0.528 0.6135 0.3010
PRICE -0.48179 1.132 -0.4256 0.678-0.122 -0.0903 -0.4722
DIST 0.50533E-01 0.4379 0.1154 0.910 0.033 0.0244 0.0105
CONSTANT 51.181 59.20 0.8646 0.404 0.242 0.0000 0.9299
|_* Regression 2
|_ols cars price
R-SQUARE = 0.1185 R-SQUARE ADJUSTED = 0.0597
VARIANCE OF THE ESTIMATE-SIGMA**2 = 9.2795
STANDARD ERROR OF THE ESTIMATE-SIGMA = 3.0462
SUM OF SQUARED ERRORS-SSE= 139.19
MEAN OF DEPENDENT VARIABLE = 6.2659
LOG OF THE LIKELIHOOD FUNCTION = -41.9944
ANALYSIS OF VARIANCE - FROM MEAN
SS DF MS F
REGRESSION 18.709 1. 18.709 2.016
ERROR 139.19 15. 9.2795 P-VALUE
TOTAL 157.90 16. 9.8688 0.176
VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 15 DF P-VALUE CORR. COEFFICIENT AT MEANS
PRICE 0.42632 0.3002 1.420 0.176 0.344 0.3442 3.6701
CONSTANT -16.730 16.21 -1.032 0.318-0.257 0.0000 -2.6701
|_* Regression 3
|_ols cars inc
R-SQUARE = 0.4551 R-SQUARE ADJUSTED = 0.4188
VARIANCE OF THE ESTIMATE-SIGMA**2 = 5.7358
STANDARD ERROR OF THE ESTIMATE-SIGMA = 2.3950
SUM OF SQUARED ERRORS-SSE= 86.037
MEAN OF DEPENDENT VARIABLE = 6.2659
LOG OF THE LIKELIHOOD FUNCTION = -37.9052
ANALYSIS OF VARIANCE - FROM MEAN
SS DF MS F
REGRESSION 71.865 1. 71.865 12.529
ERROR 86.037 15. 5.7358 P-VALUE
TOTAL 157.90 16. 9.8688 0.003
VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 15 DF P-VALUE CORR. COEFFICIENT AT MEANS
INC 0.55436E-01 0.1566E-01 3.540 0.003 0.675 0.6746 1.6539
CONSTANT -4.0973 2.985 -1.373 0.190-0.334 0.0000 -0.6539
|_test inc=20
TEST VALUE = -19.945 STD. ERROR OF TEST VALUE 0.15661E-01
T STATISTIC = -1273.4915 WITH 15 D.F. P-VALUE= 0.00000
F STATISTIC = 1621780.5 WITH 1 AND 15 D.F. P-VALUE= 0.00000
WALD CHI-SQUARE STATISTIC = 1621780.5 WITH 1 D.F. P-VALUE= 0.00000
UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 0.00000
|_test inc=0.05
TEST VALUE = 0.54357E-02 STD. ERROR OF TEST VALUE 0.15661E-01
T STATISTIC = 0.34707478 WITH 15 D.F. P-VALUE= 0.73336
F STATISTIC = 0.12046090 WITH 1 AND 15 D.F. P-VALUE= 0.73336
WALD CHI-SQUARE STATISTIC = 0.12046090 WITH 1 D.F. P-VALUE= 0.72854
UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 1.00000
|_* Regression 4
|_ols cars age
R-SQUARE = 0.5046 R-SQUARE ADJUSTED = 0.4715
VARIANCE OF THE ESTIMATE-SIGMA**2 = 5.2153
STANDARD ERROR OF THE ESTIMATE-SIGMA = 2.2837
SUM OF SQUARED ERRORS-SSE= 78.229
MEAN OF DEPENDENT VARIABLE = 6.2659
LOG OF THE LIKELIHOOD FUNCTION = -37.0966
ANALYSIS OF VARIANCE - FROM MEAN
SS DF MS F
REGRESSION 79.672 1. 79.672 15.277
ERROR 78.229 15. 5.2153 P-VALUE
TOTAL 157.90 16. 9.8688 0.001
VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 15 DF P-VALUE CORR. COEFFICIENT AT MEANS
AGE 0.16483 0.4217E-01 3.909 0.001 0.710 0.7103 1.4479
CONSTANT -2.8063 2.386 -1.176 0.258-0.291 0.0000 -0.4479
|_* Regression 5
|_ols cars dist
R-SQUARE = 0.0041 R-SQUARE ADJUSTED = -0.0623
VARIANCE OF THE ESTIMATE-SIGMA**2 = 10.484
STANDARD ERROR OF THE ESTIMATE-SIGMA = 3.2379
SUM OF SQUARED ERRORS-SSE= 157.26
MEAN OF DEPENDENT VARIABLE = 6.2659
LOG OF THE LIKELIHOOD FUNCTION = -43.0318
ANALYSIS OF VARIANCE - FROM MEAN
SS DF MS F
REGRESSION 0.64069 1. 0.64069 0.061
ERROR 157.26 15. 10.484 P-VALUE
TOTAL 157.90 16. 9.8688 0.808
VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 15 DF P-VALUE CORR. COEFFICIENT AT MEANS
DIST -0.30576E-01 0.1237 -0.2472 0.808-0.064 -0.0637 -0.0557
CONSTANT 6.6149 1.615 4.095 0.001 0.727 0.0000 1.0557
|_* Regression 6
|_ols cars inc price age dist / auxrsqr
R-SQUARE OF INC ON OTHER INDEPENDENT VARIABLES = 0.4241
R-SQUARE OF PRICE ON OTHER INDEPENDENT VARIABLES = 0.1021
R-SQUARE OF AGE ON OTHER INDEPENDENT VARIABLES = 0.3503
R-SQUARE OF DIST ON OTHER INDEPENDENT VARIABLES = 0.0645
R-SQUARE OF CONSTANT ON OTHER INDEPENDENT VARIABLES = 0.0000
R-SQUARE = 0.6527 R-SQUARE ADJUSTED = 0.5370
VARIANCE OF THE ESTIMATE-SIGMA**2 = 4.5695
STANDARD ERROR OF THE ESTIMATE-SIGMA = 2.1377
SUM OF SQUARED ERRORS-SSE= 54.835
MEAN OF DEPENDENT VARIABLE = 6.2659
LOG OF THE LIKELIHOOD FUNCTION = -34.0764
ANALYSIS OF VARIANCE - FROM MEAN
SS DF MS F
REGRESSION 103.07 4. 25.767 5.639
ERROR 54.835 12. 4.5695 P-VALUE
TOTAL 157.90 16. 9.8688 0.009
VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 12 DF P-VALUE CORR. COEFFICIENT AT MEANS
INC 0.33006E-01 0.1842E-01 1.792 0.098 0.459 0.4017 0.9847
PRICE 0.17275 0.2224 0.7769 0.452 0.219 0.1395 1.4872
AGE 0.10555 0.4897E-01 2.155 0.052 0.528 0.4549 0.9272
DIST -0.80103E-01 0.8442E-01 -0.9488 0.361-0.264 -0.1669 -0.1459
CONSTANT -14.118 11.49 -1.229 0.243-0.334 0.0000 -2.2531
|_test inc=age
TEST VALUE = -0.72544E-01 STD. ERROR OF TEST VALUE 0.61405E-01
T STATISTIC = -1.1814063 WITH 12 D.F. P-VALUE= 0.26033
F STATISTIC = 1.3957208 WITH 1 AND 12 D.F. P-VALUE= 0.26033
WALD CHI-SQUARE STATISTIC = 1.3957208 WITH 1 D.F. P-VALUE= 0.23744
UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 0.71648
|_test inc=1000*age
TEST VALUE = -105.52 STD. ERROR OF TEST VALUE 48.985
T STATISTIC = -2.1540854 WITH 12 D.F. P-VALUE= 0.05226
F STATISTIC = 4.6400838 WITH 1 AND 12 D.F. P-VALUE= 0.05226
WALD CHI-SQUARE STATISTIC = 4.6400838 WITH 1 D.F. P-VALUE= 0.03123
UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 0.21551
|_confid inc age
USING 95% AND 90% CONFIDENCE INTERVALS
CONFIDENCE INTERVALS BASED ON T-DISTRIBUTION WITH 12 D.F.
- T CRITICAL VALUES = 2.179 AND 1.782
NAME LOWER 2.5% LOWER 5% COEFFICIENT UPPER 5% UPPER 2.5% STD. ERROR
INC -0.7132E-02 0.1807E-03 0.33006E-01 0.6583E-01 0.7314E-01 0.018
AGE -0.1164E-02 0.1828E-01 0.10555 0.1928 0.2123 0.049
CONFIDENCE REGION PLOT FOR inc AND age
USING F DISTRIBUTION WITH 2 AND 12 D.F. F-VALUE = 3.890
M=MULTIPLE POINT
0.90000E-01 |
0.83684E-01 | M
0.77368E-01 | MM* ***M*
0.71053E-01 | M + MM* +
0.64737E-01 | M* *MM
0.58421E-01 | M *MM
0.52105E-01 | MM MM
0.45789E-01 | M MM
0.39474E-01 | MM *M
0.33158E-01 | M* M
0.26842E-01 | M* * M
0.20526E-01 | M* MM
0.14211E-01 | MM M
0.78947E-02 | *M* M
0.15789E-02 | MM M
-0.47368E-02 | MM M
-0.11053E-01 | + MMM + M*
-0.17368E-01 | **M*** *MM*
-0.23684E-01 | M
-0.30000E-01 |
________________________________________
-0.100 0.000 0.100 0.200 0.300
AGE
|_* Regression 7
|_ols cars inc age
R-SQUARE = 0.6076 R-SQUARE ADJUSTED = 0.5515
VARIANCE OF THE ESTIMATE-SIGMA**2 = 4.4258
STANDARD ERROR OF THE ESTIMATE-SIGMA = 2.1038
SUM OF SQUARED ERRORS-SSE= 61.961
MEAN OF DEPENDENT VARIABLE = 6.2659
LOG OF THE LIKELIHOOD FUNCTION = -35.1150
ANALYSIS OF VARIANCE - FROM MEAN
SS DF MS F
REGRESSION 95.940 2. 47.970 10.839
ERROR 61.961 14. 4.4258 P-VALUE
TOTAL 157.90 16. 9.8688 0.001
VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 14 DF P-VALUE CORR. COEFFICIENT AT MEANS
INC 0.32446E-01 0.1692E-01 1.917 0.076 0.456 0.3949 0.9680
AGE 0.11146 0.4779E-01 2.332 0.035 0.529 0.4804 0.9791
CONSTANT -5.9347 2.738 -2.168 0.048-0.501 0.0000 -0.9471
|_* Plots
|_plot cars / histo groups=5
GROUP COUNTS
GROUP 1 2 3 4 5
COUNT 0. 6. 7. 3. 1.
HISTOGRAM - CARS
PCT. N
0.529 9 I
0.471 8 I
0.471 8 I
0.471 8 I
0.412 7 I XXXXXXXXXXXX
0.412 7 I XXXXXXXXXXXX
0.412 7 I XXXXXXXXXXXX
0.353 6 I XXXXXXXXXXXXXXXXXXXXXXXX
0.353 6 I XXXXXXXXXXXXXXXXXXXXXXXX
0.353 6 I XXXXXXXXXXXXXXXXXXXXXXXX
0.294 5 I XXXXXXXXXXXXXXXXXXXXXXXX
0.294 5 I XXXXXXXXXXXXXXXXXXXXXXXX
0.294 5 I XXXXXXXXXXXXXXXXXXXXXXXX
0.235 4 I XXXXXXXXXXXXXXXXXXXXXXXX
0.235 4 I XXXXXXXXXXXXXXXXXXXXXXXX
0.235 4 I XXXXXXXXXXXXXXXXXXXXXXXX
0.176 3 I XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
0.176 3 I XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
0.176 3 I XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
0.118 2 I XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
0.118 2 I XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
0.118 2 I XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
0.059 1 I XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
0.059 1 I XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
0.059 1 I XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
I---------I---------I---------I---------I---------I---------I
-3.16 -0.171E-01 3.12 6.27 9.41 12.5 15.7
|_plot inc age
REQUIRED MEMORY IS PAR= 1 CURRENT PAR= 500
FOR MAXIMUM EFFICIENCY USE AT LEAST PAR= 2
17 OBSERVATIONS
*=INC
M=MULTIPLE POINT
247.37 | *
236.84 | *
226.32 | *
215.79 | *
205.26 | * *
194.74 | *
184.21 |
173.68 | * * * * *
163.16 | * *
152.63 |
142.11 | *
131.58 | *
121.05 |
110.53 |
100.00 | *
________________________________________
25.000 37.500 50.000 62.500 75.000
AGE
|_stop
Updated: February 18, 1998
Prepared by: Trudy Ann Cameron